Pi Found in Mathematical Calculation of the Hydrogen Atom

November 19, 2015 | Joanne Kennell

pi, decimals, circle
Photo credit: fdecomite/Flickr (CC BY 2.0)

A quantum mechanical calculation unexpectedly spits out the number.

3.14159265358979… admit it, you have tried at some point to memorize this number.  I, for one, can only ever remember to the fifth digit.  As of 2015, the representation of π is up to 13.3 trillion digits.  Why do we need to know pi up to trillions of digits?  The truth is, we don’t — it is just human curiosity and the desire to push things as far as possible.  

The number π is a strange one — it has no end and a repeating pattern never occurs.  If this number couldn’t get any weirder, a pair of physicists, Tamar Friedmann and Carl Hagan from the University of Rochester, have found that the formula for this odd number appears in a basic calculation in the physics of the hydrogen atom — the simplest atom in the universe.

SEE ALSO: Made a Poor Decision? Blame Quantum Physics.

The number π is defined as the ratio of the circumference of a circle to its diameter.  Since it is an irrational number, π cannot be expressed exactly as a fraction, and it is also a transcendental number, meaning it is not the root of any non-zero polynomial having rational coefficients.

Deriving the famous formula for π was a difficult endeavor for John Wallis, a British mathematician back in 1655.  He derived the formula for π as the product of an infinite series of ratios by first considering the ratio of the area of a circle and a square that confines it.  After pages and pages of arithmetic, he achieved the formula:

John Wallis, formula for the derivation of pi.

Much simpler ways have since been found for the derivation of π, but at its time, it was a mathematical breakthrough.

Amazingly, the famous formula from 1655 was recently found in the quantum mechanical calculation of the energy levels of a hydrogen atom.  The energy levels can be written as gamma functions, and one of those gamma functions had the value of π.  “We weren’t looking for the Wallis formula for π.  It just fell into our laps,” said Hagan, a particle physicist.

Their calculation involved a technique called the variational principle to estimate the upper limit of energy for each orbital of hydrogen.  The upper-limit estimates were compared with the exact energy values of each orbital, and the estimated energies were found to approach the exact values more closely for higher energy orbits.  The results were odd since approximations tend to work better at lower states, but that was not the case for the hydrogen atom.  “At lower energy orbits, the path of the electron is fuzzy and spread out,” explained Hagan.  “At more excited states, their orbits become more sharply defined and the uncertainty in the radius decrease.”

As exciting as it is, the discovery likely does not mean there will be a profound leap in the understanding of quantum theory, but Friedmann is glad that they were able to reveal a connection between the theory of quantum mechanics and the Wallis formula.

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